In `DeltaABC,angleA=100^(@)AD` bisects `angleA` and AD `bot` BC. Then, `angleB` is equal to

Options

A. `80^(@)`
B. `20^(@)`
C. `40^(@)`
D. `30^(@)`

Answer

To solve the problem, we need to find the measure of angle B in triangle ABC, given that angle A is 100 degrees and AD bisects angle A while being perpendicular to BC. Here are the steps to arrive at the solution:

### Step-by-Step Solution:

1. **Identify the Angles**:
- Given that angle A = 100 degrees.
- Since AD bisects angle A, we have:
- Angle BAD = 50 degrees (half of angle A)
- Angle CAD = 50 degrees (half of angle A)

2. **Recognize the Perpendicular**:
- Since AD is perpendicular to BC, we know:
- Angle ADB = 90 degrees

3. **Apply the Triangle Sum Property**:
- In triangle ABD, the sum of the angles is 180 degrees. Therefore:
\[ \text{Angle BAD} + \text{Angle ADB} + \text{Angle ABD} = 180^\circ \]
- Substituting the known values:
\[ 50^\circ + 90^\circ + \text{Angle ABD} = 180^\circ \]

4. **Calculate Angle ABD**:
- Rearranging the equation to find angle ABD:
\[ \text{Angle ABD} = 180^\circ - (50^\circ + 90^\circ) \]
\[ \text{Angle ABD} = 180^\circ - 140^\circ = 40^\circ \]

5. **Conclusion**:
- Therefore, angle B (which is angle ABD) is equal to 40 degrees.

### Final Answer:
- Angle B = 40 degrees.